Example

An insurance company determines vehicle insurance premiums based on known risk factors. If a person is considered a higher risk, their premiums will be higher. One potential factor is the color of your car. The insurance company believes that people with some color cars are more likely to get in accidents. To research this, they examine police reports for recent total-loss collisions. The data are summarized in the frequency table below which describes the frequency of each car color involved in a total-loss collision.

Color	Frequency
Blue	25
Green	52
Red	41
White	36
Black	39
Grey	23

example

Using our car data from above, note the highest frequency is 52, so our vertical axis needs to go from 0 to 52, but we might as well use 0 to 55, so that we can put a hash mark every 5 units: Notice that the height of each bar is determined by the frequency of the corresponding color.  The horizontal gridlines are a nice touch, but not necessary.  In practice, you will find it useful to draw bar graphs using graph paper, so the gridlines will already be in place, or using technology.  Instead of gridlines, we might also list the frequencies at the top of each bar, like this: The following video explains the process and value of moving data from a table to a bar graph. https://youtu.be/vwxKf_O3ui0

Example

Transforming our bar graph from earlier into a Pareto chart, we get:   The following video addresses this example of building a Pareto chart. https://youtu.be/Tsvru8DPxBE  

Example

In a survey[footnote]Gallup Poll. March 5-8, 2009. http://www.pollingreport.com/enviro.htm[/footnote], adults were asked whether they personally worried about a variety of environmental concerns. The numbers (out of 1012 surveyed) who indicated that they worried “a great deal” about some selected concerns are summarized below.  Construct a Pareto chart from the following frequency table.

Environmental Issue	Frequency
Pollution of drinking water	597
Contamination of soil and water by toxic waste	526
Air pollution	455
Global warming	354

Answer: The bar graph shown below is ordered from highest to lowest frequency and is, therefore, also a Pareto chart.

example

Once again, let's consider our vehicle color data.

Color	Frequency
Blue	25
Green	52
Red	41
White	36
Black	39
Grey	23

  We would like to create a pie chart to visualize the relative frequencies of cars in the different color categories.  To do this, we must first add up the total frequencies.  This gives us 216 cars reported to be involved in total-loss collisions. The relative frequency for each color is found by calculating the percentage of that color car involved in total-loss collision out of the number of all such cars.  The relative frequencies are included in the table below.

Color	Frequency	Relative Frequency
Blue	25	[latex]\frac{25}{216}=11.57\%[/latex]
Green	52	[latex]\frac{52}{216}=24.07\%[/latex]
Red	41	[latex]\frac{41}{216}=18.98\%[/latex]
White	36	[latex]\frac{36}{216}=16.67\%[/latex]
Black	39	[latex]\frac{39}{216}=18.06\%[/latex]
Grey	23	[latex]\frac{23}{216}=10.65\%[/latex]

  Using the relative frequencies, our pie chart might look like this:     This video demonstrates how to create pie charts like the one above. https://youtu.be/__1f8dKh6yo  

Example

Create a bar graph and a pie chart to illustrate the grades on a history exam below. A: 12 students, B: 19 students, C: 14 students, D: 4 students, F: 5 students

Answer: The following bar graph and pie chart illustrate the data described.

Example

In this example, we consider how to read a pie chart.  The pie chart below shows the percentage of voters supporting each of three candidates running for a local senate seat.  If there are 20,000 voters in the district, approximately how many support Reeves?

Answer: The pie chart shows that about 11% of the 20,000 voters in the district support Reeves.  To calculate the number of voters supporting Reeves, we compute [latex]0.11\cdot20,000\approxeq2200[/latex].  So approximately [latex]2200[/latex] voters support Reeves.

  The following video works through this example. https://youtu.be/mwa8vQnGr3I

Exercises

College students surveyed at Mountain Range University averaged 8.5 hours a day on technology: 5.5 hours of these hours were spent playing video games or on social media and 3 hours, on average, doing school work.  The survey results can be represented by the following pictogram.  

examples

Example 1: In this first example, the bar graph shown is a 3-dimensional attempt at something fancy and the distortion is likely unintentional.  Sometimes people will add features to graphs that don’t help to convey their information.  Three-dimensional graphs like this one are usually not as effective as their two-dimensional counterparts. Example 2: A labor union might produce the graph below to illustrate the difference between the average manager salary and the average worker salary. Looking at the picture, it would be reasonable to guess that the manager salaries is 4 times as large as the worker salaries – the area of the bag looks about 4 times as large. However, the manager salaries are in fact only twice as large as worker salaries, which were reflected in the picture by making the manager bag twice as tall.   The following video reviews the two examples of ineffective data representation in more detail. https://youtu.be/bFwTZNGNLKs

example

Compare the two graphs below showing support for same-sex marriage rights from a poll taken in December 2008[footnote]CNN/Opinion Research Corporation Poll. Dec 19-21, 2008, from http://www.pollingreport.com/civil.htm[/footnote]. The difference in the vertical scale on the first graph suggests a different story than the true differences in percentages, making it appear as if, in 2008, three times as many people opposed same-sex marriage rights as supported it.